Small improvements

chapter29.tex

@@ -55,7 +55,7 @@ how to divide the quadrilateral into triangles?
 It turns out that sometimes we cannot just pick
 two arbitrary opposite vertices.
 For example, in the following situation,
-the division line is outside the quadrilateral:
+the division line is \emph{outside} the quadrilateral:
 \begin{center}
 \begin{tikzpicture}[scale=0.45]
 
@@ -85,8 +85,7 @@ It is clear for a human which of the lines is the correct
 choice, but the situation is difficult for a computer.
                            
 However, it turns out that we can solve the problem using
-another method that is much easier to implement
-and does not involve any special cases.
+another method that is much easier to use.
 Namely, there is a general formula
 \[x_1y_2-x_2y_1+x_2y_3-x_3y_2+x_3y_4-x_4y_3+x_4y_1-x_1y_4,\]
 that calculates the area of a quadrilateral
@@ -132,11 +131,11 @@ following point and vector:
 The complex number class \texttt{complex} in C++ is
 useful when solving geometric problems.
 Using the class we can represent points and vectors
-as complex numbers, and the class also contains tools
+as complex numbers, and the class contains tools
 that are useful in geometry.
 
 In the following code, \texttt{C} is the type of
-a coordinate and \texttt{P} is the type of a point or vector.
+a coordinate and \texttt{P} is the type of a point or a vector.
 In addition, the code defines the macros \texttt{X} and \texttt{Y}
 that can be used to refer to x and y coordinates.
 
@@ -165,15 +164,16 @@ P s = v+u;
 cout << s.X << " " << s.Y << "\n"; // 5 3
 \end{lstlisting}
 
-An appropriate coordinate type is
+In practice,
+an appropriate coordinate type is usually
 \texttt{long long} (integer) or \texttt{long double}
-(real number), depending on the situation.
-Integers should be used whenever possible,
+(real number).
+It is a good idea to use integer whenever possible,
 because calculations with integers are exact.
 If real numbers are needed,
 precision errors should be taken into account
-when comparing them.
-A safe way to check if numbers $a$ and $b$ are equal
+when comparing numbers.
+A safe way to check if real numbers $a$ and $b$ are equal
 is to compare them using $|a-b|<\epsilon$,
 where $\epsilon$ is a small number (for example, $\epsilon=10^{-9}$).
 
@@ -182,7 +182,7 @@ where $\epsilon$ is a small number (for example, $\epsilon=10^{-9}$).
 In the following examples, the coordinate type is
 \texttt{long double}.
 
-The function \texttt{abs(v)} calculates the length
+The function $\texttt{abs}(v)$ calculates the length
 $|v|$ of a vector $v=(x,y)$
 using the formula $\sqrt{x^2+y^2}$.
 The function can also be used for
@@ -199,7 +199,7 @@ P b = {3,-1};
 cout << abs(b-a) << "\n"; // 3.16228
 \end{lstlisting}
 
-The function \texttt{arg(v)} calculates the
+The function $\texttt{arg}(v)$ calculates the
 angle of a vector $v=(x,y)$ with respect to the x axis.
 The function gives the angle in radians,
 where $r$ radians equals $180 r/\pi$ degrees.
@@ -207,7 +207,7 @@ The angle of a vector that points to the right is 0,
 and angles decrease clockwise and increase
 counterclockwise.
 
-The function \texttt{polar(s,a)} constructs a vector
+The function $\texttt{polar}(s,a)$ constructs a vector
 whose length is $s$ and that points to an angle $a$.
 In addition, a vector can be rotated by an angle $a$
 by multiplying it by a vector with length 1 and angle $a$.
@@ -228,7 +228,8 @@ cout << arg(v) << "\n"; // 0.963648
 \index{cross product}
 
 The \key{cross product} $a \times b$ of vectors
-$a=(x_1,y_1)$ and $b=(x_2,y_2)$ equals $x_1 y_2 - x_2 y_1$.
+$a=(x_1,y_1)$ and $b=(x_2,y_2)$ is calculated
+using the formula $x_1 y_2 - x_2 y_1$.
 The cross product tells us whether $b$
 turns left (positive value), does not turn (zero)
 or turns right (negative value)
@@ -265,7 +266,7 @@ The following picture illustrates the above cases:
 \end{center}
 
 \noindent
-For example, in the first picture
+For example, in the first case
 $a=(4,2)$ and $b=(1,2)$.
 The following code calculates the cross product
 using the class \texttt{complex}:
@@ -285,7 +286,7 @@ of the result is $x_1 y_2 - x_2 y_1$.
 
 \subsubsection{Point location}
 
-Cross products can be used for testing
+Cross products can be used to test
 whether a point is located on the left or right
 side of a line.
 Assume that the line goes through points
@@ -320,9 +321,9 @@ points $s_1$, $s_2$ and $p$ are on the same line.
 
 \index{line segment intersection}
 
-Consider a problem where we are given two line segments
-$ab$ and $cd$ and our task is to check whether they
-intersect. The possible cases are:
+Consider the problem of checking
+whether two given line segments
+$ab$ and $cd$ intersect. The possible cases are:
 
 \textit{Case 1:}
 The line segments are on the same line
@@ -408,7 +409,8 @@ Hence, we can use cross products to check this.
 
 \subsubsection{Point distance from a line}
 
-The area of a triangle can be calculated
+Another feature of the cross product is that
+the area of a triangle can be calculated
 using the formula
 \[\frac{| (a-c) \times (b-c) |}{2},\]
 where $a$, $b$ and $c$ are the vertices of the triangle.
@@ -438,13 +440,12 @@ it is both
 $\frac{1}{2} |s_2-s_1| d$ and
 $\frac{1}{2} ((s_1-p) \times (s_2-p))$.
 Thus, the shortest distance is
-
 \[ d = \frac{(s_1-p) \times (s_2-p)}{|s_2-s_1|} .\]
 
 \subsubsection{Point inside a polygon}
 
-Let us now consider a problem where our task is to
-find out whether a point is located inside or outside
+Let us now consider the problem of
+testing whether a point is located inside or outside
 a polygon.
 For example, in the following picture point $a$
 is inside the polygon and point $b$ is outside
@@ -463,7 +464,7 @@ the polygon.
 \end{center}
 
 A convenient way to solve the problem is to
-send a ray from the point to an arbitrary direction
+send a \emph{ray} from the point to an arbitrary direction
 and calculate the number of times it touches
 the boundary of the polygon.
 If the number is odd,
@@ -531,7 +532,7 @@ For example, the area of the polygon
 is
 \[\frac{|(2\cdot5-5\cdot4)+(5\cdot3-7\cdot5)+(7\cdot1-4\cdot3)+(4\cdot3-4\cdot1)+(4\cdot4-2\cdot3)|}{2} = 17/2.\]
 
-The idea in the formula is to go through trapezoids
+The idea of the formula is to go through trapezoids
 whose one side is a side of the polygon,
 and another side lies on the horizontal line $y=0$.
 For example:
@@ -560,12 +561,11 @@ and if $x_{i+1}<x_{i}$, the area is negative.
 
 The area of the polygon is the sum of areas of
 all such trapezoids, which yields the formula
-
 \[|\sum_{i=1}^{n-1} (x_{i+1}-x_{i}) \frac{y_i+y_{i+1}}{2}| =
 \frac{1}{2} |\sum_{i=1}^{n-1} (x_i y_{i+1} - x_{i+1} y_i)|.\]
 
 Note that the absolute value of the sum is taken,
-because the value of the sum may be positive or negative
+because the value of the sum may be positive or negative,
 depending on whether we walk clockwise or counterclockwise
 along the boundary of the polygon.
 
@@ -616,12 +616,12 @@ is $6+7/2-1=17/2$.
 \section{Distance functions}
 
 \index{distance function}
-\index{Euklidean distance}
+\index{Euclidean distance}
 \index{Manhattan distance}
 
 A \key{distance function} defines the distance between
 two points.
-The usual distance function in geometry is the
+The usual distance function is the
 \key{Euclidean distance} where the distance between
 points $(x_1,y_1)$ and $(x_2,y_2)$ is
 \[\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}.\]
@@ -723,7 +723,7 @@ between points $B$ and $C$:
 \end{tikzpicture}
 \end{center}
 
-A useful technique related to Manhattan distances
+A useful technique related to the Manhattan distance
 is to rotate all coordinates 45 degrees so that
 a point $(x,y)$ becomes $(x+y,y-x)$.
 For example, after rotating the above points,
@@ -766,7 +766,8 @@ And the maximum distance is as follows:
 
 Consider two points $p_1=(x_1,y_1)$ and $p_2=(x_1,y_1)$ whose rotated
 coordinates are $p'_1=(x'_1,y'_1)$ and $p'_2=(x'_2,y'_2)$.
-The Manhattan distance is
+Now there are two ways to express the Manhattan distance
+between $p_1$ and $p_2$:
 \[|x_1-x_2|+|y_1-y_2| = \max(|x'_1-x'_2|,|y'_1-y'_2|)\]
 
 For example, if $p_1=(1,0)$ and $p_2=(3,3)$,
@@ -775,7 +776,7 @@ and the Manhattan distance is
 \[|1-3|+|0-3| = \max(|1-6|,|-1-0|) = 5.\]
 
 The rotated coordinates provide a simple way
-to operate with Manhattan distances, because we can
+to operate with the Manhattan distance, because we can
 consider x and y coordinates separately.
 To maximize the Manhattan distance between two points,
 we should find two points whose